×

zbMATH — the first resource for mathematics

Essentially rigid floppy subgroups of the Baer-Specker group. (English) Zbl 0901.20043
The main result is that the countable direct product of the integers \(\mathbb{Z}^\omega\) contains a pure subgroup \(G\) with \(\text{End }G=\mathbb{Z}\oplus\text{Fin }G\), such that \(|\text{Hom }(G,\mathbb{Z})|=2^{\aleph_0}\) (here \(\text{Fin }G\) is the ideal of all endomorphisms of \(G\) of finite rank). This provides an answer to a question posed by Irwin, also previously solved by Blass and Göbel, under the assumption of the Continuum Hypothesis. The authors skilfully adapt already known results of others as well as of their own to prove the theorem.

MSC:
20K25 Direct sums, direct products, etc. for abelian groups
20K30 Automorphisms, homomorphisms, endomorphisms, etc. for abelian groups
03E50 Continuum hypothesis and Martin’s axiom
PDF BibTeX Cite
Full Text: DOI EuDML
References:
[1] R. Baer,Abelian groups without elements of finite order, Duke Math. J.3 (1937), 68–122. · Zbl 0016.20303
[2] G. M. Bergman,Boolean rings of projection maps, J. London Math. Soc. (2)4 (1972), 593–598. · Zbl 0229.20055
[3] A. Blass and R. Göbel,Subgroups of the Baer-Specker group with few endomorphisms but large dual, Fund. Math.149 (1996), 19–29. · Zbl 0851.20052
[4] S. U. Chase,Function topologies on abelian groups, Illinois J. Math.7 (1963), 593–608. · Zbl 0171.28703
[5] A. L. S. Corner and R. Göbel,Prescribing endomorphism algebras, a unified treatment, Proc. London Math. Soc.50 (1985), 447–479. · Zbl 0562.20030
[6] M. Dugas and R. Göbel,Endomorphism rings of separable torsion-free abelian groups, Houston J. Math11 (1985), 471–483. · Zbl 0597.20046
[7] M. Dugas, J. Irwin, and S. Khabbaz,Countable rings as endomorphism rings, J. Math. Oxford39 (1988), 201–211. · Zbl 0663.20058
[8] P. Eklof and A. Mekler,Almost Free Modules, Set-theoretic Methods, North-Holland, 1990. · Zbl 0718.20027
[9] L. Fuchs,Abelian Groups, Vol. I and II, Academic Press, 1970 and 1973. · Zbl 0213.03501
[10] R. Göbel and B. Goldsmith,On separable torsion-free modules of countable density character, J. Algebra144 (1991), 79–87. · Zbl 0737.20026
[11] R. Göbel and B. Wald,Separable torsion-free modules of small type, Houston J. Math.16 (1990), 271–288. · Zbl 0733.13009
[12] P. Hill,The additive group of commutative rings generated by idempotents, Proc. Amer. Math. Soc.38 (1973), 499–50 · Zbl 0261.20044
[13] G. Nöbeling,Verallgemeinerung eines Satzes von Herrn E. Specker, Invent. Math.6 (1968), 41–55. · Zbl 0176.29801
[14] J. Rotman,On a problem of Baer and a problem of Whitehead in abelian groups, Acta Math. Sci. Hungar.,12 (1961), 245–254. · Zbl 0101.01903
[15] E. Sąsiada,Proof that every countable and reduced torsion-free abelian group is slender, Bull. Acad. Polon. Sci.,7 (1959), 143–144. · Zbl 0085.01702
[16] S. Shelah,A combinatorial theorem and endomorphism rings of abelian groups II, Abelian Groups and Modules (R. Göbel, C. Metelli, A. Orsatti, and L. Salce, eds.), CISM Courses and Lectures 287, Springer-Verlag, 1984, pp. 37–86. · Zbl 0581.20052
[17] E. Specker,Additive Gruppen von Folgen ganzer Zahlen, Portugal. Math.9 (1950), 131–140. · Zbl 0041.36314
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.